problem solving with prime factors

Prime Factorization

Prime numbers.

A Prime Number is:

a whole number above 1 that cannot be made by multiplying other whole numbers

When it can  be made by multiplying other whole numbers it is a Composite Number , like this:

(The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19 and 23, and we have a prime number chart if you need more.)

"Factors" are the numbers you multiply together to get another number:

"Prime Factorization" is finding which prime numbers multiply together to make the original number.

Here are some examples:

Example: What are the prime factors of 12 ?

It is best to start working from the smallest prime number, which is 2, so let's check:

Yes, it divided exactly by 2. We have taken the first step!

But 6 is not a prime number, so we need to go further. Let's try 2 again:

Yes, that worked also. And 3 is a prime number, so we have the answer:

12 = 2 × 2 × 3

As you can see, every factor is a prime number , so the answer is right.

It is neater to show repeated numbers using  exponents :

• Without exponents:  2 × 2 × 3
• With exponents:  2 2 × 3

Example: What is the prime factorization of 147 ?

Can we divide 147 exactly by 2?

147 ÷ 2 = 73½

No we can't. The answer should be a whole number, and 73½ is not.

Let's try the next prime number, 3:

147 ÷ 3 = 49

That worked, now try factoring 49.

The next prime, 5, does not work. But 7 does, so we get:

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 × 7 × 7 = 3 × 7 2

Example: What is the prime factorization of 17 ?

Hang on ... 17 is a Prime Number .

So that is as far as we can go.

Another Method

We just did factorization by starting at the smallest prime and working upwards.

But sometimes it is easier to break a number down into any factors we can ... then work those factor down to primes.

Example: What are the prime factors of 90 ?

Break 90 into 9 × 10

• The prime factors of 9 are 3 and 3
• The prime factors of 10 are 2 and 5

So the prime factors of 90 are 3, 3, 2 and 5

90 = 2 × 3 2 × 5

Factor Tree

A "Factor Tree" can help: find any factors of the number, then the factors of those numbers, etc, until we can't factor any more.

Example: 48

48 = 8 × 6 , so we write down "8" and "6" below 48

Now we continue and factor 8 into 4 × 2

Then 4 into 2 × 2

And lastly 6 into 3 × 2

We can't factor any more, so we have found the prime factors.

Which reveals that 48 = 2 × 2 × 2 × 2 × 3

48 = 2 4 × 3

Why find Prime Factors?

A prime number can only be divided by 1 or itself, so it cannot be factored any further!

Every other whole number can be broken down into prime number factors.

This idea can be very useful when working with big numbers, such as in Cryptography.

Cryptography

Cryptography is the study of secret codes. Prime Factorization is important to people who try to make (or break) secret codes based on numbers.

That is because factoring very large numbers is very hard, and can take computers a long time to do.

And here is another thing:

There is only one (unique!) set of prime factors for any number.

Example: the prime factors of 330 are 2, 3, 5 and 11

330 = 2 × 3 × 5 × 11

There is no other possible set of prime numbers that can be multiplied to make 330.

This idea is so important it is called the Fundamental Theorem of Arithmetic .

Prime Factorization Tool

OK, we have one more method ... use our Prime Factorization Tool that can work out the prime factors for numbers up to 9007199254740991.

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Prime factorization using a factor tree

Once you get to the primes in your "tree", they are the "leaves", and you stop factoring in that “branch”. So 24 = 2 × 2 × 2 × 3. This is the prime factorization of 24.

 Examples:

72 has lots of factors so the factoring takes many steps.

72 = 2 × 2 × 2 × 3 × 3

We could have also started by writing 72 = 2 × 36 or 72 = 4 ×18.

1. Factor the following numbers to their prime factors.

2. Factor the following numbers to their prime factors.

By using the process above (building numbers starting from primes) you can build ANY whole number there is! Can you believe that?

We can say this in another way: ALL numbers can be factored so the factors are prime numbers. That is sort of amazing! This fact is known as the fundamental theorem of arithmetic . Indeed, it is fundamental.

See 992 factored on the right. 992 = 2 × 2 × 2 × 2 × 2 × 31. For 83,283 we get 3 × 17 × 23 × 71, and 151,282 = 2 × 3 × 3 × 3 × 11 × 11 × 23.

3. Build numbers from primes.

4. Build more numbers from primes.

5. Try it on your own! Pick 3-6 primes as you wish (you can use the same prime     several times), and see what number is built from them. 

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Prime Factorization: Definition, Methods, Examples, FAQs

What is prime factorization in math, what are factors and prime factors, how to find prime factorization of a number, solved examples on prime factorization, practice problems on prime factorization, frequently asked questions on prime factorization.

Prime factorization of a number is a way of writing a number as the product of its prime factors. Thus, the meaning of prime factorization of a number is finding prime numbers which when multiplied together give us the required number.

Example: The product prime numbers 2, 3, and 5 is 30.

Prime factorization of $30 = 2 \times 3 \times 5$

Prime Factorization Definition

Prime factorization can be defined as a way of expressing a given number as the product of its prime factors. 

If a prime number occurs more than once, we write it using exponents.

Example: Prime factorization of $18 = 2\times3\times3 = 2\times3^{2}$

Related Worksheets

A factor of a number is a number that divides the given number exactly, leaving no remainder.

Prime numbers are numbers that have only two factors, 1 and the number itself. Examples of prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, and so on.

Consider an example. The numbers 1, 2, 3, 6 are factors of the number 6.

Thus, we can say that 2 and 3 are the prime factors of 6 and express 6 as 

$6 = 2\times3$

Here, we expressed 6 as the product of its prime factors. When we write a number as a product of all its prime factors, it is called prime factorization. 

The most common methods that are used for prime factorization are:

• Prime Factorization by Factor tree method
• Prime Factorization by Division method

We will discuss these methods in detail. However, we can also find prime factors of a number by simply breaking down the number using its factors until we get prime numbers. 

Example: Find prime factorization of 48.

$42 = 8\times6$

$42 = (2\times2\times2)\times(2\times3)$

$42 = 2^{4}\times3$

Prime Factorization Methods

You can find the prime factorization of a number using various methods.

Let’s discuss the two important methods to solve prime factorization problems.

• Factor tree method
• Division method

Let us look at both the methods in detail.

Factor Tree Method

In the factor tree method, we first break down the number using the smallest prime factor of the number. The composite factors obtained in the process are further broken down until we reach the prime factors. 

The factors of the given number written in this way look like a tree. The prime numbers obtained in the end are called leaf nodes of the factor tree.

Example: Find the prime factorization of 250 using the factor tree method.

Step 1: Place the number 250 on top of the factor tree.

Step 2: Break down the given number using its smallest prime factor.

Here, $250 = 2 \times 125$

Step 3: Factorize the composite factor obtained in step 2. 

$125 = 5 \times 5$

Repeat the same step until we get all prime factors in the end. 

Thus, prime factorization of $250 = 2 \times 5^{3}$

Division Method

Step 1: Divide the given number by its smallest prime factor. 

Step 2: Divide the quotient obtained by its smallest prime factor.

Step 3: Repeat the process until the quotient becomes 1.

Step 4: Multiply all the prime factors.

Example: Prime factorization of 28 using division method.

Why Find Prime Factors?

As we already know that a prime number can only be divided by 1 or the number itself, so it cannot be factored any further. A composite number can be broken down using prime numbers and can be written uniquely as a product of its prime factors.

The prime numbers are the basic building blocks of any number.

We have many real-life uses of this concept while dealing with large numbers, such as in cryptography.

What Are the Applications of Prime Factorization?

• Finding square roots of perfect squares

It’s very easy to find square roots of perfect squares using prime factorization. 

Consider an example.

Prime factorization of $1225 = 5\times5\times7\times7$

$\sqrt{1225} = \sqrt{5\times5\times7\times7}$

$\sqrt{1225} = 5\times7$

$\sqrt{1225} = 35$

• Encryption or cryptography 
• Finding HCF and LCM using prime factorization
• Mental arithmetic
• Studying patterns

Finding HCF and LCM Using Prime Factorization

HCF (Highest Common Factor): The largest number of all the common factors of two numbers.

LCM (Least Common Multiple): The smallest number of all the common multiples of two numbers.

The HCF and LCM of two numbers can be calculated using the prime factorization method. It is quite useful when dealing with two or more large numbers. 

• First find the prime factorization of both the numbers. 
• HCF is the product of the smallest power of each common prime factor.
• LCM is the product of the greatest power of each prime factor.

Let us understand these points with the help of an example.

Example 1: Find the HCF and LCM of 42, 72, and 120 using prime factorization.

We will first find the prime factorization of the given numbers.

$42 = 2 \times 3 \times 7$

$72 = 2^{3} \times 3^{2}$

$120 = 2^{3} \times 3 \times 5$

HCF $=$ Product of the common prime factors with the lowest degree. 

HCF (42, 72, 120) $= 2 \times 3 = 6$

LCM = Product of the greatest powers of all prime factors.

LCM (42, 72, 120) $= 2^{3} \times 3^{2} \times 5 \times 7 = 2520$

Example 2: What is the HCF and LCM of 50 and 75?

The prime factorization of $50 = 2 \times 5^{2}$

The prime factorization of $75 = 3 × 5^{2}$

Multiply all the common prime factors with the lowest degree (power).

Here, we have only 5 as a common prime factor with the lowest power of 2.

Hence, HCF of (50, 75)$ = 5^{2} = 25$

LCM = Product of the greatest power of each prime factor.

Multiply all the prime factors with the highest degree.

Hence, LCM of (50, 75) $= 2^{1} \times  3^{1} \times 5^{2} = 150$

Prime Factorization of a Number

Take a look at the table showing prime factorization of some numbers.

Facts about Prime Factorization

• Cryptography is a branch of computer science that deals with methods of protecting important information using codes. Prime factorization plays an important role for the people who create a unique code, or encrypt the information using numbers that are not too heavy for computers to store and process quickly.
• 2 is the smallest even prime number. 2 is the only prime number that is even.
• 2 and 3 are the only consecutive prime numbers.
• 0 and 1 are neither prime nor composite.
• The only prime number that ends in 5 is 5.
• There is no largest prime number. The largest known prime number (as of February 2023) is $2^{82,589,933} \;−\; 1$, a number that has 24,862,048 digits when written in base 10. By the time you read this, it may be even larger.
• Mersenne prime: Prime number of the form $2^{n} \;−\; 1$, where n is a natural number.

In this article, we learned about factors, prime factors, prime factorization, methods of prime factorization, and how to find HCF and LCM using prime factorization. Let’s solve a few examples and practice problems for better understanding and revision!

1. Find the prime factorization of 24 using the factor tree method.

Let us look at the steps to find the prime factorization of 24 using the factor tree method:

Therefore, the prime factorization of $24 = 2^{3} \times 3$.

2. Find the prime factorization of 42 using the division method.

Solution: 

Divide 42 by its smallest prime factor. Next, divide the quotient by its smallest prime factor. Repeat this process until we get 1 as a quotient.

Prime factorization = Product of all prime factors

Therefore, the prime factorization of $42 = 2 \times 3 \times 7$.

3. Find the HCF of 100 and 250 using prime factorization.

To find the HCF of any number using prime factorization, we first need to find the prime factorization of both the numbers.

Prime factorization of $100 = 2^{2} \times 5^{2}$

Prime factorization of $250 = 2^{1} \times 5^{3}$

HCF (100, 250) $= 2^{1}\times 5^{2}$

Therefore, the Highest Common Factor of 100 and $250 = 50$. 

4. Find the LCM of 100 and 250 using prime factorization.

To find the LCM of any number using prime factorization, we first need to find the prime factorization of both the numbers.

LCM(100, 250) $= 2^{2} \times 5^{3}$

Therefore, the LCM of 100 and $250 = 2^{2} \times 5^{3} = 500$.

5. Write all the prime factors of 1250 using the division method.

To find the prime factors of a number, we need to find the prime factorization using one of the two methods.

Let us solve this by the division method.

Now, $\frac{1250}{2} = 625$

          $\frac{625}{5} = 125$

          $\frac{125}{5} = 25$

           $\frac{25}{5} = 5$

           $\frac{5}{5} = 1$

The prime factors of 1250 are 2 and 5. 

The prime factorization is $2 \times 5^{4}$.

Attend this quiz & Test your knowledge.

What is the prime factorization of 48?

What is the prime factorization of 15, what is the hcf of 120 and 150 using prime factorization, prime factorization is a way of expressing a given number as the ____ of its prime factors., what are the prime factors of 1000.

Is 1 a prime number?

1 is neither prime nor composite.

Is 0 prime or composite?

0 is neither prime nor composite.

How many factors does a prime number have?

Only two factors, 1 and the number itself.

Is the prime factorization of a number unique?

Yes. There is only one set of prime numbers for every whole number whose product results in the given whole number.

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Prime Factorization

Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers. Here, all the concepts of prime factors and prime factorization methods have been explained which will help the students understand how to find the prime factors of a number easily.

The simplest algorithm to find the prime factors of a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1. For example, prime factorizing the number 30 we get, 30/2 = 15, 15/3 = 5, 5/5 = 1. Since we received the remainder, it cannot be further factorized. Therefore, 30 = 2 x 3 x 5, where 2,3 and 5 are prime factors.

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19 and so on. These prime numbers when multiplied with any natural numbers produce composite numbers.

In this article, let us discuss the definition of prime factorization, different methods to find the prime factors of a number with solved examples.

What is Prime Factorization?

Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors. As we know, a composite number has more than two factors, therefore, this method is applicable only for composite numbers and not for prime numbers.

For example, the prime factors of 126 will be 2, 3 and 7 as 2 × 3 × 3 × 7 = 126 and 2, 3, 7 are prime numbers.

Prime factorization Examples

• Prime factorization of 12 is 2 × 2 × 3 = 2 2  × 3
• Prime factorization of 18 is 2 × 3 × 3 = 2 × 3 2
• Prime factorization of 24 is 2 × 2 × 2 × 3 = 2 3  × 3
• Prime factorization of 20 is 2 × 2 × 5 = 2 2  × 5
• Prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3²
• Prime Factorization of HCF and LCM

The prime numbers when multiplied by any natural numbers or whole numbers (but not 0), gives composite numbers. So basically prime factorization is performed on the composite numbers to factorize them and find the prime factors. This method is also used in the case of finding the HCF (Highest Common Factor) and LCM (Least Common Multiple) of any given set of numbers.

If any two numbers are given, then the highest common factor is the largest factor present in both the numbers whereas the least common multiple is the smallest common multiple of both the numbers.

Prime Factors of a Number

Prime factors of a number are the set of prime numbers which when multiplied by together give the actual number. Also, we can say, the prime factors divide the number completely. It is similar to factoring a number and considering only the prime numbers among the factors. For example, the prime factors of 6 will be 2 and 3, the prime factors of 26 will be 13 and 2, etc.

Prime Factorization Methods

The most commonly used prime factorization methods are:

Division Method

Factor tree method.

The steps to calculate the prime factors of a number is similar to the process of finding the factors of a large number. Follow the below steps to find the prime factors of a number using the division method:

• Step 1: Divide the given number by the smallest prime number. In this case, the smallest prime number should divide the number exactly.
• Step 2: Again, divide the quotient by the smallest prime number.
• Step 3: Repeat the process, until the quotient becomes 1.
• Step 4: Finally, multiply all the prime factors

Example of Division Method for Prime Factorization:

Below is a detailed step-by-step process of prime factorization by taking 460 as an example.

• Step 1: Divide 460 by the least prime number i.e. 2.

          So, 460 ÷ 2 = 230

• Step 2: Again Divide 230 with the least prime number (which is again 2).

          Now, 160 ÷ 2 = 115

• Step 3: Divide again with the least prime number which will be 5.

          So, 115 ÷ 5 = 23

• Step 4: As 23 is a prime number, divide it with itself to get 1.

          Now, the prime factors of 460 will be 2 2 x 5 x 23

To find the prime factorization of the given number using factor tree method, follow the below steps:

• Step 1: Consider the given number as the root of the tree
• Step 2: Write down the pair of factors as the branches of a tree
• Step 3: Again factorize the composite factors, and write down the factors pairs as the branches
• Step 4: Repeat the step, until to find the prime factors of all the composite factors

In factor tree, the factors of a number are found and then those numbers are further factorized until we reach the closure. Suppose we have to find the factors of 60 and 282 using a factor tree. Then see the diagram given below to understand the concept.

In the above figure, we can number 60 is first factorized into two numbers i.e. 6 and 10. Again, 6 and 10 is factorized to get the prime factors of 6 and 10, such that;

and 10 = 2 x 5

If we write the prime factors of 60 altogether, then;

Prime factorization of 60 = 6 x 10 = 2 x 3 x 2 x 5

Same is the case for number 282, such as;

282 = 2 x 141 = 2 x 3 x 47

So in both cases, a tree structure is formed.

Related Articles

• Prime numbers
• Factorisation
• Square Root By Prime Factorization

Prime Factorization Solved Examples

An example question is given below which will help to understand the process of calculating the prime factors of a number easily.

Q.1:  Find the prime factors of 1240.

∴ The Prime Factors of 1240 will be 2 3 × 5 × 31.

Q.2: Find the prime factors of 544.

Therefore, the prime factors of 544 are 2 5 x 17.

Prime Factorization Worksheet (Questions)

• What is the prime factorization of 48?

Write the prime factors of 2664 without using exponents.

• Is 40 = 20 × 2 an example of prime factorization process? Justify.
• Write 6393 as a product of prime factors.

Frequently Asked Questions on Prime Factorization

Define prime factorization..

Prime factorization is the process of finding the prime numbers, which are multiplied together to get the original number. For example, the prime factors of 16 are 2 × 2 × 2 × 2. This can also be written as 2 4

What are the two different methods to find the prime factors of a number?

The two different methods to find the prime factors of a number are: Division method Factor tree method

Write down the prime factorization of 13.

The prime factorization of 13 is 13. Because the prime factors of 13 are 1 and 13. As 1, and 13 are prime numbers, the prime factorization of 13 is written as 1×13, which is equal to 13.

What is the prime factorization of 999?

The prime factorization of 999 can be easily found using the factor tree method. The prime factorization of 999 is 3 3 ×37 1 , which is equal to 3×3×3×37. The numbers 3 and 37 are the prime numbers.

Find out the prime factors of 15.

The prime factors of 15 are 3×5. When the prime numbers 3 and 15 are multiplied together, we get the original number 15.

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the prime factorization of 99999

3*2 x 41*1 x 271*1

What are prime factors

Factors of a number that are prime numbers are called prime factors. These prime factors if multiplied together gives the original number. For example, prime factors of 15 is 3 and 5. 3 x 5 = 15

What is co prime number

Two or more numbers are said to be co-prime numbers, if the only common factor between them is 1. Learn more about coprime numbers with examples at BYJU’S.

what are the prime factors of 36

Prime Factorisation of 36 = 2 x 2 x 3 x 3

The prime factors of 2664 are: 2 x 2 x 2 x 3 x 3 x 37

Prime factorisation method (a)42025

Prime factorisation method for 42025: 42025 ÷ 5 = 8405 8405 ÷ 5 = 1681 1681 ÷ 41 = 41 41 ÷ 41 = 1 Thus, prime factorisation of 42025 = 5 × 5 × 41 × 41

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Repeated division

Factor tree, practice questions, prime factorization – explanation & examples.

How to find Prime Factorization?

There are two methods of finding prime factors of a number. These are repeated division and factor tree.

A number is reduced by dividing it severally with prime numbers. Prime factors of number 36 are found by repeated division as shown:

The prime factors of number 36 are, therefore, 2 and 3. This can be written as 2 × 2 × 3 × 3. It is advisable to start dividing a number by the smallest prime number and proceed to bigger factors.

What are the prime factors of 16?

The best way to solve this problem is by identifying the smallest prime factor of the number, which is 2.

Divide number by 16;

Because 8 is not a prime number, proceed by dividing again by the smallest factor;

We have the prime factors of 16 highlighted in yellow, and they include: 2 x 2 x 2 x 2.

which can be written as an exponent:

Find the prime factors of 12.

Divide 12 by 2;

6 is not prime, proceed;

Therefore, 12 = 2 x 2 x 3

12 = 2 2 × 3

It is noted that, all prime factors of a number are prime.

Factorize 147.

Start by dividing 147 by the smallest prime number.

147 ÷ 2 = 73.5

Our answer isn’t an integer, try the next prime number 3.

147 ÷ 3 = 49

Yes, 3 worked, now proceed to the next prime that can divide 49.

Therefore, 147 = 3 x 7 x 7,

= 3 x 7 2 .

What is the prime factorization of 19?

Another method on how to perform factorization is to break a number down into two integers. Now find the prime factors of the integers. This technique is useful when dealing with bigger numbers.

Find the prime factors of 210.

Break down 210 into:

210 = 21 x 10

Now calculate the factors of 21 and 10

Combine the factors: 210 = 2 x 3 x 5 x 7

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Factors, Multiple and Primes - Short Problems

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Factors And Multiples

Related Topics: More Lessons for Arithmetic Math Worksheets

If a is divisible by b , then b is a factor of a , and a is a multiple of b . For example, 30 = 3 × 10, so 3 and 10 are factors of 30 and 30 is a multiple of 3 and 10

Take note that 1 is a factor of every number.

Understanding factors and multiples is essential for solving many math problems.

Prime Factors

A factor which is a prime number is called a prime factor . For example, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5

You can use repeated division by prime numbers to obtain the prime factors of a given number.

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Prime Factorization Calculator

Welcome to our Prime Factorization Calculator page. This calculator will convert a number into its unique product of prime factors.

You can also use the calculator for finding all the factors of any given number.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Prime Factorization Calculator - How to use it

Input any number values into our Prime Factorization Calculator and it will find quickly find all of the factors and then re-write the number as a unique product of its prime factors.

It will also show you what the number looks like in exponential form.

Note: prime numbers cannot be written as a product of prime factors as their only factors are 1 and themselves (and 1 is not prime!).

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CALCULATOR RESULTS

What is prime factorization.

Prime factorization is when we split up a number into a product of its prime factors.

Every number (except prime numbers and the number 1) has its own unique set of prime factors.

To find out more about prime factorization, we hava dedicated page below.

• What is Prime Factorization support page

How is Prime Factorization useful?

Prime factorization can be useful when working out the least common multiple, or the greatest common factor.

Prime factorization is also used in cryptography (the art of writing and solving codes) to encrypt or decrypt messages.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

• Greatest Common Factor Calculator

Our Greatest Common Factor calculator will tell you the highest common factor between 2 or more numbers.

It will also list the factors of each of the numbers and tell you whether they are coprime or not.

• Least Common Multiple Calculator

Our Least Common Multiple Calculator will find the lowest common multiple of 2 or more numbers.

It will also show you the working out using a choice of two different methods.

There are also some worked examples.

Prime Factorization Worksheets

We have a collection of factor tree and prime factorization worksheets for students aged 6th grade and upwards.

Using factor trees is a great visual way of finding all the prime factors of a number.

We also have some problem solving, riddles and challenges on our Prime Factorization Worksheets page.

• Factor Tree Worksheets (easier)
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Multiples and Factors Worksheets

These sheets are all about finding multiples and factors of different numbers.

They are a great way to introduce multiples and factors and to practice this skill.

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Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

• Sieve of Eratosthenes page

Want to find out more about primes?

Take a look at our Prime Number page which clearly describes what a prime numbers is and what they are not.

There are also many different questions about prime numbers answered, as well as information about the density of primes.

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Problem-Solving Investigation: Multiples, factors & prime numbers (Year 6 Multiplication & Division)

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

4 October 2019

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Year 6 Multiplication and Division: Multiples, factors and prime numbers.

This in-depth Maths Investigation will develop maths meta-skills, and enable children to learn to think mathematically and articulate mathematical ideas.

In-depth Investigation: Magic Multiplication Squares Children complete a magic multiplication square using their knowledge of number properties and relationships. They then explore factors and multiples to create a new multiplication magic square.

This problem-solving investigation is part of our Year 6 Multiplication and Division block. Each Hamilton maths block contains a complete set of planning and resources to teach a terms worth of objectives for one of the National Curriculum for England’s maths areas.

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Year 6 Multiplication and Division - Problem-Solving Investigations

These in-depth maths investigations are open-ended problem solving activities for Year 6 children. **In-depth Investigation: Magic Multiplication Squares** Children complete a magic multiplication square using their knowledge of number properties and relationships. They then explore factors and multiples to create a new multiplication magic square. **In-depth Investigation: The Eights Have It** Children multiply numbers starting with 9 by 9 and add single-digit numbers in a decreasing sequence. They identify and describe the patterns and start to explain them. **In-depth Investigation: Awesome Answers** Using a magic square to generate 3-digit numbers, children create divisions with dividends containing specified fractions. **In-depth Investigation: Stunning Squares** Children explore patterns in the squares of numbers with reversed digits to find pairs of ‘stunning squares’. **In-depth Investigation: Geometry Genius** Children use what they know about how to find the areas of triangles and parallelograms to find the areas of rhombi, kites and trapezia. **In-depth Investigation: Get to the Root** Children use their fluency in mental multiplication to explore the patterns of digital roots in multiplication. **In-depth Investigation: Riveting Reversals** Multiply 3-digit numbers with consecutive digits by a 2-digit number; reverse the 3-digit number and repeat. Find the difference between the two answers. **In-depth Investigation: Why is it so?** Children identify a pattern in the division of a total of six numbers created using the same 3 digits. They then use algebra to explain why it is so. These investigations will develop maths meta-skills, support open-ended questioning and logical reasoning, and enable children to learn to think mathematically and articulate mathematical ideas. These problem-solving investigations come from our [Year 6 Maths Blocks](https://www.hamilton-trust.org.uk/maths/year-6-maths/). Each Hamilton maths block contains a complete set of planning and resources to teach a term’s worth of objectives for one of the National Curriculum for England’s maths areas.

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